The extended uncertainty principle inspires the Rényi entropy

Abstract: We use the extended uncertainty principle (EUP) in order to obtain the R\'{e}nyi entropy for a black hole (BH). The result implies that the non-extensivity parameter, appeared in the R\'{e}nyi entropy formalism, may be evaluated from the considerations which lead to EUP. It is also shown that, for excited BHs, the R\'{e}nyi entropy is a function of the BH principal quantum number, i.e. the BH quantum excited state. Temperature and heat capacity of the excited BHs are also investigated addressing two phases while only one of them can be stable. At this situation, whereas entropy is vanished, temperature may take a non-zero positive minimum value, depending on the value of the non-extensivity parameter. The evaporation time of excited BH has also been studied.